Useful R-norm Information Measure and its Properties
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1 IOS Jorl of Eletros Coto Eeer (IOS-JECE) e-issn: ISSN: 7-735Vole Isse (No - De 03) PP 5-57 DS oo Keert Uyy DKSr 3 Jyee Uersty of Eeer Teoloy AB o or 4736 Dstt G MP (I) Astrt : I te reset oto ew sefl -or forto esre s ee efe rterze xotlly Its rtlr ses e ee ssse Proertes of te ew esre e lso ee ste Keywors: o-tty -or etroy stost eeee tlty strto I Itroto Let s oser te set of oste rel ers ot el to eote ts y efe s : 0 Let wt s te set of ll rolty strtos P 0 for e [] ste -or forto of te strto P efe for y: P () Te -or forto esre () s rel fto efe o were s te set of rel oste ers Te esre () s fferet fro etroes of [] [3] [4] [5] Te roerty of ts esre s tt we () roes to So s etroy we P x were Te esre () e eerlze so y wys [6] Proose rterze te follow retr eerlzto of (): () P 0 0 Te oe esre () ws lle eerlze -or forto esre of eree β t rees to () we β= Frter we = () rees to: P 0 (3) I se rees to: P (4) Ts s forto esre w s ee e y [7] It e see tt (4) lso rees to So s etroy we [] Proose ste te follow retr eerlzto of (): ( P) (5) 5 Pe
2 Tey lle (5) s te eerlze -or forto esre of tye eree (5) rees to () we t frter rees to () we eetly [9] e le (5) sty te os of eerlze e oe let I orer to sts te eets E syste tke to ot we sre to e eet E E wt reset to e ltte rterst of ysl roortol to ts orte We ll te tlty or orte of eet s I eerl s eeet of (see [0]) E o- ete er E 0 retly E were rolty of orree [] rterze ttte-ltte esre w ws lle sefl forto y [0] of te exeret E s e s: lo 0 0 (6) Lter o [] rterze te follow esre of sefl forto: lo (7) Aloos to () we oser esre of sefl -or forto s e elow: () were U s te tlty strto 0 s te tlty of eet wt rolty It y e ote tt f te () rees to (7) Frter let e set of tlty strtos st U s tlty strto orreso to P I te reset er we rterze te sefl -or forto esre () xotlly seto I seto 3 we sty te roertes of te ew esre of sefl -or forto esre ' s II Axot Crterzto 3 G e seee of ftos of Let S efe oer S stsfy te follow xos: : were Axo G re o zero ostts J y0 y y: 0 y Ts xo s lso lle s roerty G stsfes te follow roerty: Axo For P U G PP : UU G P : U G P : U G P : U G P : U Axo 3 Axo 4 Let ll s otos fto of ts rets s s re erole of el tlty of eets resetely te G were 3 0 ' s Frst of ll we roe te follow tree les to fltte to roe te teore: 53 Pe
3 54 Pe Le Fro xo t s ery esy to rre t te follow ftol eto: (9) were J for Le Te otos solto tt stsfes (9) s te otos solto of te ftol eto: (0) Proof: Let e oste teers s tt Sett Fro eto (9) we e: ' () Tk () we et: () Tk () s () we e: (3) A tk () s () we et: (4) Now () toeter wt () (3) (3) rees to: (5) Ptt (5) we et te rere reslts (0) Next we ot te eerl solto of (0) Le3 Oe of te eerl otos solto of eto (0) s e y: were 0 0 (6) 0 (7) Proof: Tk (0) we e: () Te ost eerl otos solto of () (refer to [3]) s e y:
4 55 Pe 0 0 (9) 0 (0) O ssttt (9) (0) we et (6) (7) resetely Ts roes te le 3 for ll rtols 0 0 oweer y otty t ols for ll rels 0 0 Teore Te esre () e etere y te xo to 4 Proof: Ssttt te solto (6) xo we e: 0 G () Tk for e () we e: G 3 () Axo (4) toeter wt () es: It les Ptt tese les () we e G ee ts oletes te roof of teore Prtlr ses: () We tltes re ore e for e () rees to () () Frter () rees to So s etroy [3] III Proertes of sefl -or Iforto Mesre Te sefl -or forto esre stsfes te follow roertes: Proerty 3 s syetr fto of ter rets roe tt te ertto of s s re tke toeter Proerty 3 Proof: for
5 56 Pe Tk Proerty 33 Ato of two eets wose rolty of orree s zero or tlty s zero s o effet o sefl forto e 0 0 Proof: Let s oser Slrly we roe tt Proerty 34 stsfes te o-tty of te follow for: V Q U P were Q P V U Proof: S= I I I V Q U P = LS Proerty 35 Let A A e two eets roltes tltes resetely te we efe te tlty of te oo eet A A s: A A (3) Teore 3 Uer te oosto lw (3) te follow ols: were Proof:
6 Ts oletes te roof of teore 3 IV Colso - or forto esre s efe rterze we te rolty strto P elo to - or etor se Ts s ew to to te fly of eerlze forto esres I reset er we e osere tt ysl syste s ltte rterzto to to ttte e efe rterze ew esre -or forto esre Ts esre frter e eerlze y wys e le sore o we sore syols e tlty lso to to rolty of orree eferees [] D E Boekee JCA V Der Le Te -or Iforto esre Iforto otrol [] CE So A tetl teory of oto Bell Syste Teoloy Jorl [3] A ey O esre of etroy forto Selete er of Alfore ey [4] Jr F Crt Qtfto of lssfto roesses oet of Strtrl - etroy Kyeretk [5] Z Drozy Geerlze forto fto Iforto otrol [6] DS oo At Crterzto of te eerlze -or etroy Cre orl of Mtetl Coter see 00-3 [7] SAroto Iforto teoretl oserto o estto roes Iforto otrol [] DS oo Dk Sr Geerlze -or forto esres Jorl of Alto Mt sttsts & Iforts 4() [9] DS oo Sxe Sol Dk Sr A Geerlze -or etroy o teore Itertol orl of Mtetl Sees Eeer Altos [0] G Loo Qttte-Qltte Mesre of Iforto Srer-Verl NewYork 97 [] M Bels Gs A ttte ltte esre of forto Cyerets Syste IEEE Trs Iforto Teory IT [] US Bker DS oo Me le rterzto of sefl forto esres Tk Jorl of Mtets [3] J Azel Letres o ftol etos ter ltos (Ae Press New York 966) 57 Pe
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